Paper analyzes online PCA, proving nearly optimal error bound.
problem Theoretical convergence analysis of online PCA.
method Stochastic approximation iteration for streaming data.
result Proves nearly optimal finite-sample error bound for online PCA.
Paper explores stochastic algorithms for PCA, focusing on convergence and runtime.
problem Finding top k eigen vectors of data covariance matrix.
method Revisits and analyzes stochastic approaches to PCA optimization.
result Stochastic methods offer comparable or superior empirical performance to direct non-convex methods.
The paper uses diffusion approximations to analyze and optimize online principal component estimation.
problem Optimizing online principal component estimation from streaming data.
method Diffusion approximation tools applied to Oja's iteration for principal component analysis.
result The Oja's iteration for the top eigenvector generates a continuous-state discrete-time Markov chain over the unit sphere.
Lazy SPCA simplifies SPCA for large datasets with similar performance.
problem Efficiently reducing high-dimensional datasets for large-scale computations.
method Derives a simplified algorithm (Lazy SPCA) with reduced computational complexity.
result Lazy SPCA finds the same principal subspace as SPCA and maintains similar pairwise distances.
A new method reduces data movement in neural network training.
problem Large data movement during neural network training.
method Streaming batch principal component analysis for low-rank updates.
result Effective training of convolutional neural networks with low overhead.
A novel Riemannian extension of stochastic variance reduction for manifold optimization.
problem Optimization on the Grassmann manifold for large-scale problems.
method Riemannian stochastic variance reduced gradient (R-SVRG) on the Grassmann manifold.
result The proposed algorithm outperforms standard Riemannian SGD on various problems.
Efficient private matrix analysis algorithms for recent variants.
problem Private analysis of recent matrix updates.
method Identifying sufficient conditions on positive semidefinite matrices.
result First efficient differentially private algorithms for various matrix analysis tasks.
A novel online framework for analyzing multidimensional functional data.
problem Analysis of multidimensional functional data streams poses significant challenges.
method Online functional principal component analysis using tensor product splines on a Stiefel manifold with Riemannian stochastic gradient descent.
result Efficient and scalable modeling of multidimensional functional data.
Proposes flexible auto-encoders for varying data dimensions.
problem Fixed latent dimensions limit data flexibility.
method Stochastic bottleneck with weighted dropouts.
result Seamless variable dimensionality reduction with high performance.
The paper introduces a method for interpretable principal component analysis of high-dimensional time series.
problem Inconsistent and difficult-to-interpret principal component estimates in high-dimensional regimes.
method Localized sparse principal component analysis of spectral density matrices in frequency domain.
result Efficient algorithm for sparse-localized estimates of principal subspaces.
We propose a new two stage algorithm LING for large scale regression problems. LING has the same risk as the well known Ridge Regression under the fixed design setting and can be computed much faster. Our experiments have shown that LING performs well in terms of both prediction accuracy and computational efficiency co…
Modularity component analysis clusters data without centering.
problem Clustering data without centering.
method Developed exact linear relation between modularity matrix eigenvectors and singular vectors.
result Modularity component analysis clusters data similarly to PCA but without centering.
Generalizes PCA to maximize any convex function of components.
problem Finding a principal vector that maximizes a convex function of components.
method Gradient ascent algorithm for solving the generalized PCA problem; fixed points of neural networks for kernel version.
result Solutions can be obtained as fixed points of simple neural networks.
Analyzes last few principal components for stock correlations.
problem Identifying highly correlated stocks for better portfolio management.
method Principal component analysis of correlation matrix.
result Last few components contain useful financial information.
In this work, we develop a novel principal component analysis (PCA) for semimartingales by introducing a suitable spectral analysis for the quadratic variation operator. Motivated by high-dimensional complex systems typically found in interest rate markets, we investigate correlation in high-dimensional high-frequency …
Two new PCA variants improve financial data analysis.
problem Numerical instability and nonstationarity in PCA for finance.
method Iterated and exponentially weighted moving PCA variants using Ogita-Aishima iteration.
result Improved stability and adaptability in financial data analysis.
New simulations advise caution in choosing principal components for multivariate functional data.
problem Inaccurate selection of principal components in multivariate functional data.
method Extensive simulations investigating the reliability of percentage of variance explained thresholds.
result Conventional threshold methods may fail to accurately explain overall variance in multivariate functional data.
Essential principal components simplify spectral analysis with minimal training data.
problem Accurate spectral quantification from complex mixtures.
method Identifying essential principal components and using molar extinction coefficients.
result Near one-to-one projection from principal components to mixture constituents.
Study on dynamics of non-linear autoencoders learning principal components.
problem Technical difficulty in studying non-linear autoencoders due to non-trivial correlations.
method Derive asymptotically exact equations for SGD training of shallow, non-linear autoencoders.
result Autoencoders learn principal components sequentially and tie weights are ineffective.
In this dissertation, the main goal is visualisation of financial time series. We expect that visualisation of financial time series will be a useful auxiliary for technical analysis. Firstly, we review the technical analysis methods and test our trading rules, which are built by the essential concepts of technical ana…
QAPCA uses quantum annealing for robust PCA.
problem Outliers in data skew L2-norm principal components.
method Quantum annealing for L1-norm optimization.
result QAPCA's reconstruction error is comparable to L1-BF.
Develops an ℓ_p theory for PCA and spectral clustering.
problem Lack of precise characterizations of PCA scores for low-dimensional embedding.
method An ℓ_p perturbation theory for PCA in Hilbert spaces, analyzing eigenvectors and Gram matrix.
result Optimal recovery results for Gaussian mixture and stochastic block models.
Paper uses PCA to analyze Chinese sovereign bonds and discusses bond immunization.
problem Analyzing factors affecting Chinese sovereign bond yield changes.
method Applied Principal Component Analysis (PCA) on bond yield data.
result Identified principal factors influencing Chinese sovereign bond yield changes.
Efficiently projects vectors onto top PCA components without explicit PCA.
problem Efficiently project vectors onto top principal components of a matrix.
method Iterative algorithm using ridge regression and polynomial approximation.
result First runtime improvement for principal component regression.
New Gaussian DPP model reveals directionality in data.
problem Negative dependence in data modeling.
method Investigation of Gaussian Determinantal Processes (GDPs) with parametric modulation.
result Parameter modulation introduces directionality in repulsion structure, affecting dependency.
Robust PCA method works under uncertain covariance.
problem Principal component analysis under uncertain covariance.
method Robust streaming PCA with temporal uncertainty set.
result Noisy power method is rate-optimal in our setting.
Two novel clustering methods improve community detection in networks.
problem Community detection in networks using principal components.
method Principal Component Clustering (PCC) and Normalized Principal Component Clustering (NPCC).
result NPCC provides significant improvement over PCC and RSC methods.
We study sparse principal component analysis for high dimensional vector autoregressive time series under a doubly asymptotic framework, which allows the dimension d to scale with the series length T. We treat the transition matrix of time series as a nuisance parameter and directly apply sparse principal component…
Proposes an online method for high-dimensional streaming data.
problem Increasing variable dimensions with sample size in online kernel sliced inverse regression.
method Introduces approximate linear dependence condition and dictionary variable sets to address the problem. Transforms into online generalized eigen-decomposition problem and uses stochastic optimization for updates.
result Achieves close performance to batch processing kernel sliced inverse regression.
We describe and analyze a simple algorithm for principal component analysis and singular value decomposition, VR-PCA, which uses computationally cheap stochastic iterations, yet converges exponentially fast to the optimal solution. In contrast, existing algorithms suffer either from slow convergence, or computationally…
GT-PCA improves PCA for image and time series data.
problem Lack of robustness to transformations in PCA.
method GT-PCA is a neural network that estimates components invariant to specific transformations.
result GT-PCA outperforms alternative methods in synthetic and real data experiments.
Study explores K-means clustering of variables and its relation to PCA.
problem Exploring the relationship between K-means clustering of variables and PCA.
method Apply PCA to original data and K-means to transposed data, quantify variable contributions to principal components.
result Identifies how variable clusters contribute to principal components identified by PCA.
Develops robust methods for infinite-dimensional stochastic processes.
problem Measuring covariations in stochastic evolution equations in infinite dimensions.
method Asymptotic theory for jump robust measurement of covariations.
result Identifies scaling limits for realized covariations.
Paper develops a dual formulation for PCA in Hilbert spaces.
problem Characterizing probabilistic PCA in Hilbert spaces.
method Dual formulation for probabilistic PCA in Hilbert spaces.
result Generative framework for kernel methods developed.
Paper extends principal component pursuit to hypercomplex numbers for improved audio data analysis.
problem Improving robust principal component analysis for audio data.
method Extends principal component pursuit to polar n-complex and n-bicomplex numbers, deriving proximity operators for ℓ1- and trace-norm regularizers. result Our approach outperforms tensor robust principal component analysis on audio data.
The paper uses PCA and HMM to forecast stock returns outperforming buy-and-hold.
problem Predicting stock returns accurately.
method Applied PCA to covariance matrix of S&P 500 stocks, used HMM on principal components, and forecasted stock returns.
result The model outperforms buy-and-hold strategy in terms of annualized Sharpe ratio.
A new method uses Gram matrix for efficient multivariate functional principal components.
problem Efficiently estimating eigencomponents of multidimensional functional datasets.
method Proposes using inner-product matrix to estimate eigenelements of multivariate and multidimensional functional datasets.
result Established relationship between eigenelements of covariance operator and inner-product matrix.
Develops statistical framework for analyzing functional data extremes.
problem Analyzing extremes of functional data in Hilbert spaces.
method Regular variation in Hilbert spaces, Peaks-Over-Threshold framework, functional PCA.
result Proposes a dimension reduction method for functional extreme observations.
DPCA reveals dynamic patterns of multiple air pollutants in Houston.
problem Identifying the mixture of air pollutants in a region with dynamic chemistry and transport.
method Dynamic Principal Component Analysis (DPCA) applied to normalized multivariate time series of daily pollutant measurements.
result DPCA captures time-dependent correlation structure of pollutants and explains up to 90% variability in ambient air data.
Explains eigenvalue and generalized eigenvalue problems with examples.
problem Eigenvalue and generalized eigenvalue problems.
method Introduction and examples from machine learning.
result Solutions to eigenvalue and generalized eigenvalue problems.
Two derivations of PCA for distributional data.
problem PCA for datasets of distributions.
method Two derivations: variance maximization and reconstruction error minimization.
result Closed-form solution for distributional PCA.
This research simplifies PCA model selection using MDL principle.
problem Choosing the right number of principal components in PCA.
method Reduces NML problems to lower-dimension problems and bounds PCA NML.
result Bound the NML of PCA by terms of the NML of linear regression.
Improved convergence speed of principal component analysis through modified learning rules.
problem Slow convergence for covariance matrices with close eigenvalues.
method Introduced an additional term to the objective function to mitigate convergence issues.
result Significantly improved convergence speed confirmed through simulations.
Improved data analysis with robust SPCA algorithm.
problem Identifying localized spatial structures and disambiguating time scales in low-rank data.
method Formulated as a value-function optimization problem, then extended with randomized linear algebra methods for scalability.
result Robust and efficient sparse principal components in corrupted data.
New techniques solve robust principal component analysis problems.
problem Robust Principal Component Analysis (RPCA) problems.
method Dual smoothing and level set techniques in convex optimization.
result Numerous theoretical and practical improvements for RPCA.
A method for integrating multiple cancer data sources using kernel principal component analysis.
problem Lack of comprehensive analysis of cancer subtypes from multiple data sources.
method Unsupervised data integration method based on kernel principal component analysis with a scoring function to determine input matrix impact.
result Enables visualization and clustering of integrated data for cancer subtype identification.
Improved estimation of multiple principal components using manifold optimization and iterative deflation techniques.
problem Estimating multiple principal components efficiently and orthogonally.
method Extended SFPCA using manifold optimization and iterative deflation techniques.
result Alternative deflation schemes improve signal extraction and component estimation.
We propose a fair principal component analysis method that balances reconstruction error and subgroup fairness.
problem Fairness and robustness in principal component analysis for consequential domains.
method Distributionally robust optimization over the Stiefel manifold with a Riemannian subgradient descent.
result The proposed method achieves better performance on real-world datasets compared to state-of-the-art baselines.